Interpreting Low and High Order Rules: A Granular Computing Approach
|
|
- Lucas Cain
- 5 years ago
- Views:
Transcription
1 Interpreting Low and High Order Rules: A Granular Computing Approach Yiyu Yao, Bing Zhou and Yaohua Chen Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 {yyao, zhou200b, chen115y}@cs.uregina.ca Abstract. The main objective of this paper is to provide a granular computing based interpretation of rules representing two levels of knowledge. This is done by adopting and adapting the decision logic language for granular computing. The language provides a formal method for describing and interpreting conditions in rules as granules and rules as relationships between granules. An information table is used to construct a concrete granular computing model. Two types of granules are constructed from an information table. They lead to two types of rules called low order and high order rules. As examples, we examine rules in the standard rough set analysis and dominance-based rough set analysis. Keywords. Low order rules, high order rules, granular computing, dominancebased rough set analysis 1 Introduction Rules are a commonly used form for representing knowledge. Two levels of knowledge may be expressed in terms of low and high order rules, respectively [16]. A low order rule expresses connections between attribute values of the same object. Classification rules are a typical example of low order rules. For example, a classification rule may state that if the Hair color is blond and Eye color is blue, then the Class is +. A high order rule expresses connections of different objects in terms of their attribute values. Functional dependencies are a typical example of high order rules. For example, a functional dependency rule may state that if two persons have the same Hair color, then they will have the same Eye color. The notion of a high order rule is also related to relational learning, in which a k-ary predicate is used to define a relation between k objects [10]. For simplicity, in this paper we only consider binary relations. In rough set analysis [6 8], a decision logic language DL is used to build conditions in low order rules and to interpret these conditions as subsets (i.e., granules) of objects. This language is referred to as L 0 [16]. In order to describe high order rules, an extended language L 1 is introduced [16]. In L 1, conditions are interpreted in terms of a set of object pairs. The two languages share the same syntactic rules, with two semantic interpretations. That is, their main differences lie in the different interpretations of atomic formulas. It is therefore possible to introduce a common language.
2 In this paper, we propose a decision logic language L for granular computing. Instead of expressing the atomic formulas by a particular concrete type of conditions, we treat them as primitive notions that can be interpreted differently. This flexibility enables us to describe different types of rules. The language is interpreted in the Tarski s style through the notion of a model and satisfiability. The model is a non-empty domain consisting of a set of individuals. An individual satisfies a formula if the individual has the properties as specified by the formula. A concept is therefore jointly defined by a pair consisting of the intension of the concept, a formula of the language, and the extension of the concept, a subset of the model. As illustrative examples to show the usefulness of the proposed language, we analyze rules in the standard rough set analysis [6 8] and dominance-based rough set analysis [2 4, 11, 15]. 2 A Decision Logic Language for Granular Computing By extracting the high-level similarities of the decision logic languages DL, L 0, and L 1, we propose a logic language L for granular computing. The language L is constructed from a finite set of atomic formulas, denoted by A = {p, q,...}. Each atomic formula may be interpreted as representing one piece of basic knowledge. The physical meaning of atomic formulas becomes clearer in a particular application. In general, an atomic formula corresponds to one particular property of an individual under discussion. The construction of atomic formulas is an essential step of knowledge representation. The set of atomic formulas provides a basis on which more complex knowledge can be represented. Compound formulas can be built recursively from atomic formulas by using logic connectives. If φ and ψ are formulas, then so are ( φ), (φ ψ), (φ ψ), (φ ψ), and (φ ψ). The semantics of the language L can be defined in the Tarski s style through the notion of a model and satisfiability. The model is a nonempty domain consisting of a set of individuals, denoted by M = {x, y,...}. The meaning of formulas can be given recursively. For an atomic formula p, we assume that an individual x M either satisfies p or does not satisfy p, but not both. For an individual x M, if it satisfies an atomic formula p, we write x = p, otherwise, we write x p. The satisfiability of an atomic formula by individuals of M is viewed to be the basic knowledge describable by the language L. An individual satisfies a formula if the individual has the properties as specified by the formula. Let φ and ψ be two formulas, the satisfiability of compound formulas is defined as follows: (1). x = φ iff x φ, (2). x = φ ψ iff x = φ and x = ψ, (3). x = φ ψ iff x = φ or x = ψ, (4). x = φ ψ iff x = φ ψ, (5). x = φ ψ iff x = φ ψ and x = ψ φ. In order to emphases the roles played by the set of atomic formulas A and the set of individuals M, we also rewrite the language L as L(A, M).
3 The construction of the set of atomic formulas and the model M depends on a particular application. For modeling different problems, we may choose different sets of atomic formulas and models. The language L therefore is flexible and enables us to describe a variety of problems. With the notion of satisfiability, one can introduce a set-theoretic interpretation of formulas of the language L. If φ is a formula, the meaning of φ in the model M is the set of individuals defined by [7]: m(φ) = {x M x = φ}. (1) That is, m(φ) is the set of individuals satisfying a formula φ. This enables us to establish a correspondence between logic connectives and set-theoretic operators. Specifically, the following properties hold: (C1). (C2). (C3). (C4). (C5). m( φ) = m(φ), m(φ ψ) = m(φ) m(ψ), m(φ ψ) = m(φ) m(ψ), m(φ ψ) = m(φ) m(ψ), m(φ ψ) = (m(φ) m(ψ)) ( m(φ) m(ψ)), where m(φ) = M m(φ) denotes the set complement of m(φ). In the study of concepts, many interpretations have been proposed and examined. The classical view regards a concept as a unit of thought consisting of two parts, i.e., the intension and extension of the concept [1, 13]. By using the language L, we obtain a simple and precise representation of a concept in terms of its intension and extension. That is, a concept is defined by a pair (m(φ), φ). The formula φ is the description of properties shared by individuals in m(φ), and m(φ) is the set of individuals satisfying φ. A concept is thus described jointly by its intension and extension. This formulation enables us to study concepts in a logic setting in terms of intensions and in a set-theoretic setting in terms of extensions. The language L provides a formal method for describing granules. Elements of a granule may be interpreted as instances of a concept, i.e., the extension of the concept. The formula is a formal description of the granule. In this way, the language L only enables us to define certain subsets of M. For an arbitrary subset of M, we may not be able to construct a formula for it. In other words, depending on the set of atomic formulas, the language L can only describe a restricted family of subsets of M. 3 Interpretation of Low and High Order Rules Using the Language L We interpret different types of rules of an information table as concrete applications to show the usefulness of the language L.
4 3.1 Information Table An information table provides a convenient way to describe a finite set of objects by a finite set of attributes [7]. Formally, an information table can be expressed as: where S = (U, At, {V a a At}, {{R a } a At}, {I a a At}), U is a finite nonempty set of objects called universe, At is a finite nonempty set of attributes, V a is a nonempty set of values for a At, {R a } is a family of binary relations on V a, I a : U V a is an information function. Each information function I a maps an object in U to a value of V a for an attribute a At. Our definition of an information table considers more knowledge and information about relationships between values of attributes. Each relation R a can represent similarity, dissimilarity, or ordering of values in V a [1]. The equality relation = is only a special case of R a. The rough set theory and the DL language use the trivial equality relation on attribute values [7]. Pawlak and Skowron [8] consider a more generalized notion of an information table. For each attribute a At, a relational structure R a over V a is introduced. Furthermore, a language can be defined based on the relational structures. A binary relation is a special case of relational structures. Thus, the discussion of this paper may be viewed as a special case of Pawlak and Skowron s formulation. 3.2 Low Order Rules For interpreting low order rules, we construct a language by using U as the model M. That is, individuals of M are objects in the universe U. The set of atomic formulas are constructed as follows. With respect to an attribute a At and an attribute value v V a, an atomic formula of the language L is denoted by (a, R a, v). An object x U satisfies an atomic formula (a, R a, v) if the value of x on attribute a is R a -related to the value v, that is I a (x) R a v, we write: x = (a, R a, v) iff I a (x) R a v. We denote the language for interpreting low order rules as L({(a, R a, v)}, U). The granule corresponding to the atomic formula (a, R a, v), namely, its meaning set, is defined as: m(a, R a, v) = {x U I a (x)r a v}. Granules corresponding to compound formulas are defined by Equation (1).
5 A low order rule can be derived according to the relationships between these granules. We can express rules in the form φ ψ by using formulas of the language L. For easy understanding, we reexpress the formula (a, R a, v) in another form based on the definition of satisfiability of the atomic formulas. An example of a low order rule is given as: Low Order rule: (I ai (x) R ai v ai ) (I dj (x) R dj v dj ), where x U, v ai V ai, v dj V dj, a i At, and d j At. For simplicity, we only use conjunction in the rule. 3.3 High Order Rules For interpreting high order rules, we construct a language by using U U as the model M. That is, individuals of M are object pairs in U U. The set of atomic formulas are constructed as follows. With respect to an attribute a At, an atomic formula of the language L is denoted by (a, R a ). A pair of objects (x, y) U U satisfies an atomic formula (a, R a ) if the value of x is R a -related to the value of y on the attribute a, that is, I a (x) R a I a (y). We write: (x, y) = (a, R a ) iff I a (x) R a I a (y). For clarity, we denote the language as L({(a, R a )}, U U). The granule corresponding to the atomic formula (a, R a ), i.e., the meaning set, is defined as: m(a, R a ) = {(x, y) U U I a (x)r a I a (y)}. Granules corresponding to the compound formulas are defined by Equation (1). A high order rule expresses the relationships between these granules. An example of a high order rule is given as: High Order rule: (I ai (x) R ai I ai (y)) (I dj (x) R dj I dj (y)), where (x, y) U U, a i At, d j At. 3.4 Quantitative Measures of Rules The meanings and interpretations of a rule φ ψ can be further clarified by using the extensions m(φ) and m(ψ) of the two concepts. More specifically, we can define quantitative measures indicating the strength of a rule. A systematic analysis of probabilistic quantitative measures can be found in [14]. Two examples of probabilistic quantitative measures are [12]: accuracy(φ ψ) = m(φ ψ), coverage(φ ψ) = m(φ) m(φ ψ), (2) m(ψ)
6 Table 1. Rough Set Approaches for Studying Low and High Order Rules Relation Low Order Rule High Order Rule Method Generalized R I a (x)r a v a I d (x)r d v d I a (x)r a I a (y) I d (x)r d I d (y) Rough Set Analysis Standard = I a (x) = v a I d (x) = v d I a (x) = I a (y) I d (x) = I d (y) Rough Set Analysis Dominance-based I a(x) a v a I d (x) d v d I a(x) a I a(y) I d (x) d I d (y) Rough Set Analysis where denotes the cardinality of a set. The two measures are applicable to both low and high order rules. This demonstrates the flexibility and power of the language L. While the accuracy reflects the correctness of the rule, the coverage reflects the applicability of the rule. If accuracy(φ ψ) = 1, we have a strong association between φ and ψ. A smaller value of accuracy indicates a weak association. A higher coverage suggests that the relationships of more individuals can be derived from the rule. The accuracy and coverage are not independent of each other, one may observe a trade-off between accuracy and coverage. A rule with higher coverage may have a lower accuracy, while a rule with higher accuracy may have a lower coverage. 4 Rough Set Approaches on Rules In this section, we use two rough set approaches [2 4, 6 8, 11, 15] as examples to illustrate the usefulness of the language L. The basic results are summarized in Table 1. For comparison, we also include the results of generalized rough set analysis based on an arbitrary binary relation R a on attribute values. 4.1 Standard Rough Set Analysis The standard rough set analysis is based on the trivial equality relation on attribute values [6 8]. It is used for the extraction of rules for classification and attribute dependency analysis. By using the language L, the standard rough set approach can be formulated as follows. For low order rules, the language is given by L({(a, =, v)}, U) with atomic formulas of the form of (a, =, v). An object x U satisfies an atomic formula (a, =, v) if the value of x on attribute a is v, that is, I a (x) = v. We write: x = (a, =, v) iff I a (x) = v. The granule corresponding to the atomic formula (a, =, v) is: m(a, =, v) = {x U I a (x) = v}. The granule m(a, =, v) is also referred to as the block defined by the attribute-value pair (a, v) [5]. Blocks correspond to the atomic formulas and are used to construct
7 Table 2. An Information Table Object Height Hair Eyes Class o 1 short blond blue + o 2 short blond brown - o 3 tall red blue + o 4 tall dark blue - o 5 tall dark blue - o 6 tall blond blue + o 7 tall dark brown - o 8 short blond brown - rules. Low order rules can be expressed based on the equality relation =. An example of a low order rule is: (I ai (x) = v ai ) (I dj (x) = v dj ), where x U, v ai V ai, v dj V dj, a i At, and d j At. For high order rules, the language is given by L({(a, =)}, U U) with atomic formulas of the form of (a, =). A pair of objects (x, y) U U satisfies an atomic formula (a, =) if the value of x equals to the value of y on attribute a, that is, I a (x) = I a (y). We write: (x, y) = (a, =) iff I a (x) = I a (y). The granule corresponding to the atomic formula (a, =) is: m(a, =) = {(x, y) U U I a (x) = I a (y)}. High order rules can be expressed by using the equality relation =. An example of a high order rule is: (I ai (x) = I ai (y)) (I dj (x) = I dj (y)), where (x, y) U U, a i At, d j At. Example 1. Table 2 is an information table taken from [9]. Each object is described by four attributes. The column labeled by Class denotes an expert s classification of the objects. An example of a low order rule in this information table is: LR 1 : (I Hair (x) = blond) (I Eyes (x) = blue) (I Class (x) = +). That is, if an object has blond hair and blue eyes, then it belongs to class +. An example of a high order rule is: HR 1 : (I Height (x) = I Height (y)) (I Eyes (x) = I Eyes (y)) (I Class (x) = I Class (y)).
8 That is, if two objects have the same height and the same eye color, then they belong to the same class. By using the probabilistic quantitative measures, we have: accuracy(lr 1 ) = 1, coverage(lr 1 ) = 2/3. The association between (Hair, =, blond) (Eyes, =, blue) and (Class, =, +) reaches the maximum value 1, and the applicability of the rule is also high. For rule HR 1, we have: accuracy(hr 1 ) = 7/11, coverage(hr 1 ) = 7/17. In this case, (Height, =) (Eyes, =) does not tell us too much information about the overall objects classification in terms of both accuracy and coverage. 4.2 Dominance-based Rough Set Analysis The dominance-based rough set analysis proposed by Greco, Matarazzo and Slowinski [2 4] is based on preference relations on attribute values. It is used for the extraction of rules for ranking and attribute dependency analysis. Several different types of rules are introduced in dominance-based rough set analysis. In what follows, we interpret two types of such rules by using the language L, as demonstrated in [11, 15, 16]. For low order rules, the language is given by L({(a, a, v)}, U). The granule corresponding to the atomic formula (a, a, v) is defined as: m(a, a, v) = {x U I a (x) a v}. Low order rules can be expressed by using preference relations. An example of a low order rule is: (I ai (x) ai v ai ) (I dj (x) dj v dj ), where x U, v ai V ai, v dj V dj, a i At, and d j At. For high order rules, the language is given by L({(a, a )}, U U). The granule corresponding to the atomic formula (a, a ) is defined as: m(a, a ) = {(x, y) U U I a (x) a I a (y)}. High order rules can also be expressed by using the preference relations. An example of a high order rule is: (I ai (x) ai I ai (y)) (I dj (x) dj I dj (y)). where (x, y) U U, a i At, d j At.
9 Table 3. An Information Table with Preference Relations Objects Size Warranty Price Weight Overall p 1 middle 3 years $200 heavy 1 p 2 large 3 years $300 very heavy 3 p 3 small 3 years $300 light 3 p 4 small 3 years $250 very light 2 p 5 small 2 years $200 very light 3 Example 2. Table 3, taken from [11], is an information table with preference relations on attribute values. It is a group of five products by five manufactures, each product is described by four attributes. The final ranking labeled by Overall may be determined by their market share of the products. The preference relations induce the following orderings: Size : small Size middle Size large, Warranty : 3 years Warranty 2 years, Price : $200 Price $250 Price $300, Weight : very light Weight light Weight heavy Weight very heavy, Overall : 1 Overall 2 Overall 3. An example of a low order rule in this information table is: LR 2 : (I Size (x) middle) (I Warranty (x) 3 years) I Overall (x) 2. That is, if a product s size is greater than or equal to middle and warranty is greater than or equal to 3 years, then its overall ranking will be greater than or equal to 2. An example of a high order rule is: HR 2 : (I Size (x) I Size (y)) (I Price (x) I Price (y)) I Overall (x) I Overall (y). That is, if one product s size is smaller than or the same as another product and the price is not higher, then this product s overall ranking will be greater than or equal to the other product. By using the quantitative measures, we have: accuracy(lr 2 ) = 2/3, coverage(lr 2 ) = 1. There exists a strong association between the two concepts, and applicability of the rule reaches the highest level. Similarly, for rule HR 2, we have: accuracy(hr 2 ) = 11/13, coverage(hr 2 ) = 11/18. The concept (Size, ) (Price, ) reflects the overall objects ranking positively in terms of both accuracy and coverage.
10 5 Conclusion A granular computing based interpretation is presented in this paper. By extracting the high-level similarity from existing decision logic languages [7, 16], we introduce a more general language L for granular computing. Two basic features of the language L are the set of atomic formulas A and a model M consisting of individuals. For each formula, the collection of all individuals satisfying the formula form a granule, called the meaning of the formula. A rule is therefore expressed as connection between two formulas and interpreted based on the corresponding granules of the two formulas. Depending on particular applications, we can construct concrete languages by using different types of atomic formulas and the associated models. This flexibility of the language L is demonstrated by considering two rough set approaches, namely, the standard rough set analysis and dominance-based rough set analysis. The main differences of the two approaches are their respective treatments of atomic formulas and models. An information table is used to construct a concrete granular computing model. For standard rough set analysis, two types of granules are constructed based on two families of atomic formulas. One consists of a set of objects that share the same attribute value. The other consists of object pairs that cannot be distinguished based on the values of an attribute. Low and high order rules are defined to describe relationships between these two types of granules. For dominance-based rough set analysis, similar interpretations can be obtained by using two different families of atomic formulas. The results of the paper suggest that one may study rule mining at a more abstract level. Algorithms and evaluation measures can indeed be designed uniformly for both low and high order rules. References 1. Demri, S., Orlowska, E.: Logical Analysis of Indiscernibility. In: Orlowska, E. (Ed.): Incomplete Information: Rough Set Analysis. Physica Verlag, Heidelberg (1997) Greco, S., Matarazzo, B., Slowinski, R.: Rough Approximation of a Preference Relation by Dominance Relations. European Journal of Operational Research 117 (1999) Greco, S., Matarazzo, B., Slowinski, R.: Rough Approximation by Dominance Relations. International Journal of Intelligent Systems 17 (2002) Greco, S., Slowinski, R., Stefanowski, J.: Mining Association Rules in Preference-ordered Data. Proceedings of the 13th International Symposium on Foundations of Intelligent Systems (ISMIS 02) (2002) Grzymala-Busse, J.W.: Incomplete Data and Generalization of Indiscernibility Relation, Definability, and Approximations. Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Proceedings of 10th International Conference, LNAI 3641 (2005) Nguyen, H.S., Skowron, A., Stepaniuk, J.: Granular Computing: a Rough Set Approach. Computational Intelligence 17 (2001) Pawlak, Z.: Rough Sets - Theoretical Aspects of Reasoning about Data. Kluwer Publishers, Boston. (1991) 8. Pawlak, Z., Skowron, A.: Rough Sets: Some Extensions. Information Science 177 (2007) 28-40
11 9. Quinlan, J.R.: Learning Efficient Classification Procedures and Their Application to Chess End-games. In: Michalski, J.S., Carbonell, J. G., Mirchell, T.M. (Eds.): Machine Learning: An Artificial Intelligence Approach. Vol. 1. Morgan Kaufmann. (1983) Quinlan, J.R.: Learning Logical Definitions from Relations. Machine Learning 5 (1990) Sai, Y., Yao, Y.Y., Zhong, N.: Data Analysis and Mining in Ordered Information Tables. Proceedings of the 2001 IEEE International Conference on Data Mining. (2001) Tsumoto, S. Antomated Discovery of Plausible Rules Based on Rough Sets and Rough Inclusion. Proceedings of PAKDD 99, LNAI 1574 (1999) Wille, R.: Concept Lattices and Conceptual Knowledge Systems. Computers Mathematics with Applications 23 (1992) Yao, Y.Y., Zhong, N.: An Analysis of Quantitative Measures Associated with Rules. Proceedings of PAKDD 99, LNAI 1574 (1999) Yao, Y.Y., Sai, Y.: On Mining Ordering Rules. New Frontiers in Artificial Intelligence, Joint JSAI 2001 Workshop Post-Proceedings, LNCS 2253 (2001) Yao, Y.Y.: Mining High Order Decision Rules. In: Inuiguchi, M., Hirano, S., and Tsumoto, S. (Eds.): Rough Set Theory and Granular Computing. Springer, Berlin. (2003)
Concept Lattices in Rough Set Theory
Concept Lattices in Rough Set Theory Y.Y. Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca URL: http://www.cs.uregina/ yyao Abstract
More informationA Generalized Decision Logic in Interval-set-valued Information Tables
A Generalized Decision Logic in Interval-set-valued Information Tables Y.Y. Yao 1 and Qing Liu 2 1 Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca
More informationFoundations of Classification
Foundations of Classification J. T. Yao Y. Y. Yao and Y. Zhao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 {jtyao, yyao, yanzhao}@cs.uregina.ca Summary. Classification
More informationClassification Based on Logical Concept Analysis
Classification Based on Logical Concept Analysis Yan Zhao and Yiyu Yao Department of Computer Science, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 E-mail: {yanzhao, yyao}@cs.uregina.ca Abstract.
More informationNaive Bayesian Rough Sets
Naive Bayesian Rough Sets Yiyu Yao and Bing Zhou Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 {yyao,zhou200b}@cs.uregina.ca Abstract. A naive Bayesian classifier
More informationNotes on Rough Set Approximations and Associated Measures
Notes on Rough Set Approximations and Associated Measures Yiyu Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca URL: http://www.cs.uregina.ca/
More informationA Logical Formulation of the Granular Data Model
2008 IEEE International Conference on Data Mining Workshops A Logical Formulation of the Granular Data Model Tuan-Fang Fan Department of Computer Science and Information Engineering National Penghu University
More informationDrawing Conclusions from Data The Rough Set Way
Drawing Conclusions from Data The Rough et Way Zdzisław Pawlak Institute of Theoretical and Applied Informatics, Polish Academy of ciences, ul Bałtycka 5, 44 000 Gliwice, Poland In the rough set theory
More informationREDUCTS AND ROUGH SET ANALYSIS
REDUCTS AND ROUGH SET ANALYSIS A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN COMPUTER SCIENCE UNIVERSITY
More informationRough Sets for Uncertainty Reasoning
Rough Sets for Uncertainty Reasoning S.K.M. Wong 1 and C.J. Butz 2 1 Department of Computer Science, University of Regina, Regina, Canada, S4S 0A2, wong@cs.uregina.ca 2 School of Information Technology
More informationMinimal Attribute Space Bias for Attribute Reduction
Minimal Attribute Space Bias for Attribute Reduction Fan Min, Xianghui Du, Hang Qiu, and Qihe Liu School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu
More informationAction Rule Extraction From A Decision Table : ARED
Action Rule Extraction From A Decision Table : ARED Seunghyun Im 1 and Zbigniew Ras 2,3 1 University of Pittsburgh at Johnstown, Department of Computer Science Johnstown, PA. 15904, USA 2 University of
More informationSimilarity-based Classification with Dominance-based Decision Rules
Similarity-based Classification with Dominance-based Decision Rules Marcin Szeląg, Salvatore Greco 2,3, Roman Słowiński,4 Institute of Computing Science, Poznań University of Technology, 60-965 Poznań,
More informationRough Set Model Selection for Practical Decision Making
Rough Set Model Selection for Practical Decision Making Joseph P. Herbert JingTao Yao Department of Computer Science University of Regina Regina, Saskatchewan, Canada, S4S 0A2 {herbertj, jtyao}@cs.uregina.ca
More informationEasy Categorization of Attributes in Decision Tables Based on Basic Binary Discernibility Matrix
Easy Categorization of Attributes in Decision Tables Based on Basic Binary Discernibility Matrix Manuel S. Lazo-Cortés 1, José Francisco Martínez-Trinidad 1, Jesús Ariel Carrasco-Ochoa 1, and Guillermo
More informationThree Discretization Methods for Rule Induction
Three Discretization Methods for Rule Induction Jerzy W. Grzymala-Busse, 1, Jerzy Stefanowski 2 1 Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, Kansas 66045
More informationRoman Słowiński. Rough or/and Fuzzy Handling of Uncertainty?
Roman Słowiński Rough or/and Fuzzy Handling of Uncertainty? 1 Rough sets and fuzzy sets (Dubois & Prade, 1991) Rough sets have often been compared to fuzzy sets, sometimes with a view to introduce them
More informationSome remarks on conflict analysis
European Journal of Operational Research 166 (2005) 649 654 www.elsevier.com/locate/dsw Some remarks on conflict analysis Zdzisław Pawlak Warsaw School of Information Technology, ul. Newelska 6, 01 447
More informationA new Approach to Drawing Conclusions from Data A Rough Set Perspective
Motto: Let the data speak for themselves R.A. Fisher A new Approach to Drawing Conclusions from Data A Rough et Perspective Zdzisław Pawlak Institute for Theoretical and Applied Informatics Polish Academy
More informationComparison of Rough-set and Interval-set Models for Uncertain Reasoning
Yao, Y.Y. and Li, X. Comparison of rough-set and interval-set models for uncertain reasoning Fundamenta Informaticae, Vol. 27, No. 2-3, pp. 289-298, 1996. Comparison of Rough-set and Interval-set Models
More informationThe size of decision table can be understood in terms of both cardinality of A, denoted by card (A), and the number of equivalence classes of IND (A),
Attribute Set Decomposition of Decision Tables Dominik Slezak Warsaw University Banacha 2, 02-097 Warsaw Phone: +48 (22) 658-34-49 Fax: +48 (22) 658-34-48 Email: slezak@alfa.mimuw.edu.pl ABSTRACT: Approach
More informationA Scientometrics Study of Rough Sets in Three Decades
A Scientometrics Study of Rough Sets in Three Decades JingTao Yao and Yan Zhang Department of Computer Science University of Regina [jtyao, zhang83y]@cs.uregina.ca Oct. 8, 2013 J. T. Yao & Y. Zhang A Scientometrics
More informationLevel Construction of Decision Trees in a Partition-based Framework for Classification
Level Construction of Decision Trees in a Partition-base Framework for Classification Y.Y. Yao, Y. Zhao an J.T. Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canaa S4S
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 59 (2010) 431 436 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A short
More informationROUGH set methodology has been witnessed great success
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 2, APRIL 2006 191 Fuzzy Probabilistic Approximation Spaces and Their Information Measures Qinghua Hu, Daren Yu, Zongxia Xie, and Jinfu Liu Abstract Rough
More informationFeature Selection with Fuzzy Decision Reducts
Feature Selection with Fuzzy Decision Reducts Chris Cornelis 1, Germán Hurtado Martín 1,2, Richard Jensen 3, and Dominik Ślȩzak4 1 Dept. of Mathematics and Computer Science, Ghent University, Gent, Belgium
More informationEnsembles of classifiers based on approximate reducts
Fundamenta Informaticae 34 (2014) 1 10 1 IOS Press Ensembles of classifiers based on approximate reducts Jakub Wróblewski Polish-Japanese Institute of Information Technology and Institute of Mathematics,
More informationRough operations on Boolean algebras
Rough operations on Boolean algebras Guilin Qi and Weiru Liu School of Computer Science, Queen s University Belfast Belfast, BT7 1NN, UK Abstract In this paper, we introduce two pairs of rough operations
More informationCRITERIA REDUCTION OF SET-VALUED ORDERED DECISION SYSTEM BASED ON APPROXIMATION QUALITY
International Journal of Innovative omputing, Information and ontrol II International c 2013 ISSN 1349-4198 Volume 9, Number 6, June 2013 pp. 2393 2404 RITERI REDUTION OF SET-VLUED ORDERED DEISION SYSTEM
More informationTolerance Approximation Spaces. Andrzej Skowron. Institute of Mathematics. Warsaw University. Banacha 2, Warsaw, Poland
Tolerance Approximation Spaces Andrzej Skowron Institute of Mathematics Warsaw University Banacha 2, 02-097 Warsaw, Poland e-mail: skowron@mimuw.edu.pl Jaroslaw Stepaniuk Institute of Computer Science
More informationRough Set Approaches for Discovery of Rules and Attribute Dependencies
Rough Set Approaches for Discovery of Rules and Attribute Dependencies Wojciech Ziarko Department of Computer Science University of Regina Regina, SK, S4S 0A2 Canada Abstract The article presents an elementary
More informationAn algorithm for induction of decision rules consistent with the dominance principle
An algorithm for induction of decision rules consistent with the dominance principle Salvatore Greco 1, Benedetto Matarazzo 1, Roman Slowinski 2, Jerzy Stefanowski 2 1 Faculty of Economics, University
More informationData Analysis - the Rough Sets Perspective
Data Analysis - the Rough ets Perspective Zdzisław Pawlak Institute of Computer cience Warsaw University of Technology 00-665 Warsaw, Nowowiejska 15/19 Abstract: Rough set theory is a new mathematical
More informationTwo Semantic Issues in a Probabilistic Rough Set Model
Fundamenta Informaticae 108 (2011) 249 265 249 IOS Press Two Semantic Issues in a Probabilistic Rough Set Model Yiyu Yao Department of Computer Science University of Regina Regina, Canada yyao@cs.uregina.ca
More informationOn the Relation of Probability, Fuzziness, Rough and Evidence Theory
On the Relation of Probability, Fuzziness, Rough and Evidence Theory Rolly Intan Petra Christian University Department of Informatics Engineering Surabaya, Indonesia rintan@petra.ac.id Abstract. Since
More informationSelected Algorithms of Machine Learning from Examples
Fundamenta Informaticae 18 (1993), 193 207 Selected Algorithms of Machine Learning from Examples Jerzy W. GRZYMALA-BUSSE Department of Computer Science, University of Kansas Lawrence, KS 66045, U. S. A.
More informationAction rules mining. 1 Introduction. Angelina A. Tzacheva 1 and Zbigniew W. Raś 1,2,
Action rules mining Angelina A. Tzacheva 1 and Zbigniew W. Raś 1,2, 1 UNC-Charlotte, Computer Science Dept., Charlotte, NC 28223, USA 2 Polish Academy of Sciences, Institute of Computer Science, Ordona
More informationA PRIMER ON ROUGH SETS:
A PRIMER ON ROUGH SETS: A NEW APPROACH TO DRAWING CONCLUSIONS FROM DATA Zdzisław Pawlak ABSTRACT Rough set theory is a new mathematical approach to vague and uncertain data analysis. This Article explains
More informationClassification of Voice Signals through Mining Unique Episodes in Temporal Information Systems: A Rough Set Approach
Classification of Voice Signals through Mining Unique Episodes in Temporal Information Systems: A Rough Set Approach Krzysztof Pancerz, Wies law Paja, Mariusz Wrzesień, and Jan Warcho l 1 University of
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationROUGH SET THEORY FOR INTELLIGENT INDUSTRIAL APPLICATIONS
ROUGH SET THEORY FOR INTELLIGENT INDUSTRIAL APPLICATIONS Zdzisław Pawlak Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Poland, e-mail: zpw@ii.pw.edu.pl ABSTRACT Application
More informationOn Probability of Matching in Probability Based Rough Set Definitions
2013 IEEE International Conference on Systems, Man, and Cybernetics On Probability of Matching in Probability Based Rough Set Definitions Do Van Nguyen, Koichi Yamada, and Muneyuki Unehara Department of
More informationModeling the Real World for Data Mining: Granular Computing Approach
Modeling the Real World for Data Mining: Granular Computing Approach T. Y. Lin Department of Mathematics and Computer Science San Jose State University, San Jose, California 95192-0103 and Berkeley Initiative
More informationLeast Common Subsumers and Most Specific Concepts in a Description Logic with Existential Restrictions and Terminological Cycles*
Least Common Subsumers and Most Specific Concepts in a Description Logic with Existential Restrictions and Terminological Cycles* Franz Baader Theoretical Computer Science TU Dresden D-01062 Dresden, Germany
More informationAPPLICATION FOR LOGICAL EXPRESSION PROCESSING
APPLICATION FOR LOGICAL EXPRESSION PROCESSING Marcin Michalak, Michał Dubiel, Jolanta Urbanek Institute of Informatics, Silesian University of Technology, Gliwice, Poland Marcin.Michalak@polsl.pl ABSTRACT
More informationRough Sets, Rough Relations and Rough Functions. Zdzislaw Pawlak. Warsaw University of Technology. ul. Nowowiejska 15/19, Warsaw, Poland.
Rough Sets, Rough Relations and Rough Functions Zdzislaw Pawlak Institute of Computer Science Warsaw University of Technology ul. Nowowiejska 15/19, 00 665 Warsaw, Poland and Institute of Theoretical and
More informationSemantic Rendering of Data Tables: Multivalued Information Systems Revisited
Semantic Rendering of Data Tables: Multivalued Information Systems Revisited Marcin Wolski 1 and Anna Gomolińska 2 1 Maria Curie-Skłodowska University, Department of Logic and Cognitive Science, Pl. Marii
More informationAn Introduction to Description Logics
An Introduction to Description Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, 21.11.2013 Marco Cerami (UPOL) Description Logics 21.11.2013
More informationOn the Structure of Rough Approximations
On the Structure of Rough Approximations (Extended Abstract) Jouni Järvinen Turku Centre for Computer Science (TUCS) Lemminkäisenkatu 14 A, FIN-20520 Turku, Finland jjarvine@cs.utu.fi Abstract. We study
More informationDecision tables and decision spaces
Abstract Decision tables and decision spaces Z. Pawlak 1 Abstract. In this paper an Euclidean space, called a decision space is associated with ever decision table. This can be viewed as a generalization
More informationFuzzy Modal Like Approximation Operations Based on Residuated Lattices
Fuzzy Modal Like Approximation Operations Based on Residuated Lattices Anna Maria Radzikowska Faculty of Mathematics and Information Science Warsaw University of Technology Plac Politechniki 1, 00 661
More informationAndrzej Skowron, Zbigniew Suraj (Eds.) To the Memory of Professor Zdzisław Pawlak
Andrzej Skowron, Zbigniew Suraj (Eds.) ROUGH SETS AND INTELLIGENT SYSTEMS To the Memory of Professor Zdzisław Pawlak Vol. 1 SPIN Springer s internal project number, if known Springer Berlin Heidelberg
More informationBulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 67-82
Bulletin of the Transilvania University of Braşov Vol 10(59), No. 1-2017 Series III: Mathematics, Informatics, Physics, 67-82 IDEALS OF A COMMUTATIVE ROUGH SEMIRING V. M. CHANDRASEKARAN 3, A. MANIMARAN
More informationGranular Computing: Granular Classifiers and Missing Values
1 Granular Computing: Granular Classifiers and Missing Values Lech Polkowski 1,2 and Piotr Artiemjew 2 Polish-Japanese Institute of Information Technology 1 Koszykowa str. 86, 02008 Warsaw, Poland; Department
More informationBrock University. Probabilistic granule analysis. Department of Computer Science. Ivo Düntsch & Günther Gediga Technical Report # CS May 2008
Brock University Department of Computer Science Probabilistic granule analysis Ivo Düntsch & Günther Gediga Technical Report # CS-08-04 May 2008 Brock University Department of Computer Science St. Catharines,
More informationOn rule acquisition in incomplete multi-scale decision tables
*Manuscript (including abstract) Click here to view linked References On rule acquisition in incomplete multi-scale decision tables Wei-Zhi Wu a,b,, Yuhua Qian c, Tong-Jun Li a,b, Shen-Ming Gu a,b a School
More informationLearning Rules from Very Large Databases Using Rough Multisets
Learning Rules from Very Large Databases Using Rough Multisets Chien-Chung Chan Department of Computer Science University of Akron Akron, OH 44325-4003 chan@cs.uakron.edu Abstract. This paper presents
More informationMathematical Approach to Vagueness
International Mathematical Forum, 2, 2007, no. 33, 1617-1623 Mathematical Approach to Vagueness Angel Garrido Departamento de Matematicas Fundamentales Facultad de Ciencias de la UNED Senda del Rey, 9,
More informationBanacha Warszawa Poland s:
Chapter 12 Rough Sets and Rough Logic: A KDD Perspective Zdzis law Pawlak 1, Lech Polkowski 2, and Andrzej Skowron 3 1 Institute of Theoretical and Applied Informatics Polish Academy of Sciences Ba ltycka
More informationOn Improving the k-means Algorithm to Classify Unclassified Patterns
On Improving the k-means Algorithm to Classify Unclassified Patterns Mohamed M. Rizk 1, Safar Mohamed Safar Alghamdi 2 1 Mathematics & Statistics Department, Faculty of Science, Taif University, Taif,
More informationAn Analysis of Quantitative Measures Associated with Rules
An Analysis of Quantitative Measures Associated with Rules Y.Y. Yao 1 and Ning Zhong 2 1 Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca
More informationRough sets, decision algorithms and Bayes' theorem
European Journal of Operational Research 136 2002) 181±189 www.elsevier.com/locate/dsw Computing, Arti cial Intelligence and Information Technology Rough sets, decision algorithms and Bayes' theorem Zdzisøaw
More informationOn flexible database querying via extensions to fuzzy sets
On flexible database querying via extensions to fuzzy sets Guy de Tré, Rita de Caluwe Computer Science Laboratory Ghent University Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium {guy.detre,rita.decaluwe}@ugent.be
More informationOn Granular Rough Computing: Factoring Classifiers through Granulated Decision Systems
On Granular Rough Computing: Factoring Classifiers through Granulated Decision Systems Lech Polkowski 1,2, Piotr Artiemjew 2 Department of Mathematics and Computer Science University of Warmia and Mazury
More informationKnowledge Discovery. Zbigniew W. Ras. Polish Academy of Sciences, Dept. of Comp. Science, Warsaw, Poland
Handling Queries in Incomplete CKBS through Knowledge Discovery Zbigniew W. Ras University of orth Carolina, Dept. of Comp. Science, Charlotte,.C. 28223, USA Polish Academy of Sciences, Dept. of Comp.
More informationAijun An and Nick Cercone. Department of Computer Science, University of Waterloo. methods in a context of learning classication rules.
Discretization of Continuous Attributes for Learning Classication Rules Aijun An and Nick Cercone Department of Computer Science, University of Waterloo Waterloo, Ontario N2L 3G1 Canada Abstract. We present
More informationAbstract. In this paper we present a query answering system for solving non-standard
Answering Non-Standard Queries in Distributed Knowledge-Based Systems Zbigniew W. Ras University of North Carolina Department of Comp. Science Charlotte, N.C. 28223, USA ras@uncc.edu Abstract In this paper
More information1 Introduction Rough sets theory has been developed since Pawlak's seminal work [6] (see also [7]) as a tool enabling to classify objects which are on
On the extension of rough sets under incomplete information Jerzy Stefanowski 1 and Alexis Tsouki as 2 1 Institute of Computing Science, Poznań University oftechnology, 3A Piotrowo, 60-965 Poznań, Poland,
More informationInformation Refinement and Revision for Medical Expert System Automated Extraction of Hierarchical Rules from Clinical Data
From: AAAI Technical Report SS-02-03. Compilation copyright 2002, AAAI (www.aaai.org). All rights reserved. Information Refinement and Revision for Medical Expert System Automated Extraction of Hierarchical
More informationHierarchical Structures on Multigranulation Spaces
Yang XB, Qian YH, Yang JY. Hierarchical structures on multigranulation spaces. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 27(6): 1169 1183 Nov. 2012. DOI 10.1007/s11390-012-1294-0 Hierarchical Structures
More informationComputational Intelligence, Volume, Number, VAGUENES AND UNCERTAINTY: A ROUGH SET PERSPECTIVE. Zdzislaw Pawlak
Computational Intelligence, Volume, Number, VAGUENES AND UNCERTAINTY: A ROUGH SET PERSPECTIVE Zdzislaw Pawlak Institute of Computer Science, Warsaw Technical University, ul. Nowowiejska 15/19,00 665 Warsaw,
More informationRough Sets and Conflict Analysis
Rough Sets and Conflict Analysis Zdzis law Pawlak and Andrzej Skowron 1 Institute of Mathematics, Warsaw University Banacha 2, 02-097 Warsaw, Poland skowron@mimuw.edu.pl Commemorating the life and work
More informationARTICLE IN PRESS. Information Sciences xxx (2016) xxx xxx. Contents lists available at ScienceDirect. Information Sciences
Information Sciences xxx (2016) xxx xxx Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins Three-way cognitive concept learning via multi-granularity
More informationSets with Partial Memberships A Rough Set View of Fuzzy Sets
Sets with Partial Memberships A Rough Set View of Fuzzy Sets T. Y. Lin Department of Mathematics and Computer Science San Jose State University San Jose, California 95192-0103 E-mail: tylin@cs.sjsu.edu
More informationResearch Article Special Approach to Near Set Theory
Mathematical Problems in Engineering Volume 2011, Article ID 168501, 10 pages doi:10.1155/2011/168501 Research Article Special Approach to Near Set Theory M. E. Abd El-Monsef, 1 H. M. Abu-Donia, 2 and
More informationInterval based Uncertain Reasoning using Fuzzy and Rough Sets
Interval based Uncertain Reasoning using Fuzzy and Rough Sets Y.Y. Yao Jian Wang Department of Computer Science Lakehead University Thunder Bay, Ontario Canada P7B 5E1 Abstract This paper examines two
More informationGranularity, Multi-valued Logic, Bayes Theorem and Rough Sets
Granularity, Multi-valued Logic, Bayes Theorem and Rough Sets Zdzis law Pawlak Institute for Theoretical and Applied Informatics Polish Academy of Sciences ul. Ba ltycka 5, 44 000 Gliwice, Poland e-mail:zpw@ii.pw.edu.pl
More informationML techniques. symbolic techniques different types of representation value attribute representation representation of the first order
MACHINE LEARNING Definition 1: Learning is constructing or modifying representations of what is being experienced [Michalski 1986], p. 10 Definition 2: Learning denotes changes in the system That are adaptive
More informationA Rough Set Interpretation of User s Web Behavior: A Comparison with Information Theoretic Measure
A Rough et Interpretation of User s Web Behavior: A Comparison with Information Theoretic easure George V. eghabghab Roane tate Dept of Computer cience Technology Oak Ridge, TN, 37830 gmeghab@hotmail.com
More informationAN INTRODUCTION TO FUZZY SOFT TOPOLOGICAL SPACES
Hacettepe Journal of Mathematics and Statistics Volume 43 (2) (2014), 193 204 AN INTRODUCTION TO FUZZY SOFT TOPOLOGICAL SPACES Abdülkadir Aygünoǧlu Vildan Çetkin Halis Aygün Abstract The aim of this study
More informationNeighborhoods Systems: Measure, Probability and Belief Functions
202-207. Neighborhoods Systems: Measure, Probability and Belief Functions T. Y. Lin 1 * tylin@cs.sjsu.edu Y, Y, Yao 2 yyao@flash.lakeheadu.ca 1 Berkeley Initiative in Soft Computing, Department of Electrical
More informationRestricted role-value-maps in a description logic with existential restrictions and terminological cycles
Restricted role-value-maps in a description logic with existential restrictions and terminological cycles Franz Baader Theoretical Computer Science, Dresden University of Technology D-01062 Dresden, Germany
More informationOutlier Detection Using Rough Set Theory
Outlier Detection Using Rough Set Theory Feng Jiang 1,2, Yuefei Sui 1, and Cungen Cao 1 1 Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences,
More informationBayes Theorem - the Rough Set Perspective
Bayes Theorem - the Rough Set Perspective Zdzis law Pawlak Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul. Ba ltycka 5, 44 100 Gliwice, Poland MOTTO: I had come to an
More information2 WANG Jue, CUI Jia et al. Vol.16 no", the discernibility matrix is only a new kind of learning method. Otherwise, we have to provide the specificatio
Vol.16 No.1 J. Comput. Sci. & Technol. Jan. 2001 Investigation on AQ11, ID3 and the Principle of Discernibility Matrix WANG Jue (Ξ ±), CUI Jia ( ) and ZHAO Kai (Π Λ) Institute of Automation, The Chinese
More informationCompleting Description Logic Knowledge Bases using Formal Concept Analysis
Completing Description Logic Knowledge Bases using Formal Concept Analysis Franz Baader 1, Bernhard Ganter 1, Ulrike Sattler 2 and Barış Sertkaya 1 1 TU Dresden, Germany 2 The University of Manchester,
More informationROUGHNESS IN MODULES BY USING THE NOTION OF REFERENCE POINTS
Iranian Journal of Fuzzy Systems Vol. 10, No. 6, (2013) pp. 109-124 109 ROUGHNESS IN MODULES BY USING THE NOTION OF REFERENCE POINTS B. DAVVAZ AND A. MALEKZADEH Abstract. A module over a ring is a general
More informationPartial Approximative Set Theory: A Generalization of the Rough Set Theory
International Journal of Computer Information Systems and Industrial Management Applications. ISSN 2150-7988 Volume 4 (2012) pp. 437-444 c MIR Labs, www.mirlabs.net/ijcisim/index.html Partial Approximative
More informationMining Approximative Descriptions of Sets Using Rough Sets
Mining Approximative Descriptions of Sets Using Rough Sets Dan A. Simovici University of Massachusetts Boston, Dept. of Computer Science, 100 Morrissey Blvd. Boston, Massachusetts, 02125 USA dsim@cs.umb.edu
More informationNEAR GROUPS ON NEARNESS APPROXIMATION SPACES
Hacettepe Journal of Mathematics and Statistics Volume 414) 2012), 545 558 NEAR GROUPS ON NEARNESS APPROXIMATION SPACES Ebubekir İnan and Mehmet Ali Öztürk Received 26:08:2011 : Accepted 29:11:2011 Abstract
More informationFUZZY PARTITIONS II: BELIEF FUNCTIONS A Probabilistic View T. Y. Lin
FUZZY PARTITIONS II: BELIEF FUNCTIONS A Probabilistic View T. Y. Lin Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192, USA tylin@cs.sjsu.edu 1 Introduction
More informationRepresenting Uncertain Concepts in Rough Description Logics via Contextual Indiscernibility Relations
Representing Uncertain Concepts in Rough Description Logics via Contextual Indiscernibility Relations Nicola Fanizzi 1, Claudia d Amato 1, Floriana Esposito 1, and Thomas Lukasiewicz 2,3 1 LACAM Dipartimento
More informationDynamic Programming Approach for Construction of Association Rule Systems
Dynamic Programming Approach for Construction of Association Rule Systems Fawaz Alsolami 1, Talha Amin 1, Igor Chikalov 1, Mikhail Moshkov 1, and Beata Zielosko 2 1 Computer, Electrical and Mathematical
More informationCharacterizing Pawlak s Approximation Operators
Characterizing Pawlak s Approximation Operators Victor W. Marek Department of Computer Science University of Kentucky Lexington, KY 40506-0046, USA To the memory of Zdzisław Pawlak, in recognition of his
More informationResearch on Complete Algorithms for Minimal Attribute Reduction
Research on Complete Algorithms for Minimal Attribute Reduction Jie Zhou, Duoqian Miao, Qinrong Feng, and Lijun Sun Department of Computer Science and Technology, Tongji University Shanghai, P.R. China,
More informationResearch Article Decision Analysis via Granulation Based on General Binary Relation
International Mathematics and Mathematical Sciences Volume 2007, Article ID 12714, 13 pages doi:10.1155/2007/12714 esearch Article Decision Analysis via Granulation Based on General Binary elation M. M.
More informationHigh Frequency Rough Set Model based on Database Systems
High Frequency Rough Set Model based on Database Systems Kartik Vaithyanathan kvaithya@gmail.com T.Y.Lin Department of Computer Science San Jose State University San Jose, CA 94403, USA tylin@cs.sjsu.edu
More informationApplied Logic. Lecture 3 part 1 - Fuzzy logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw
Applied Logic Lecture 3 part 1 - Fuzzy logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018 1
More informationChapter 18 Rough Neurons: Petri Net Models and Applications
Chapter 18 Rough Neurons: Petri Net Models and Applications James F. Peters, 1 Sheela Ramanna, 1 Zbigniew Suraj, 2 Maciej Borkowski 1 1 University of Manitoba, Winnipeg, Manitoba R3T 5V6, Canada jfpeters,
More informationFUZZY ASSOCIATION RULES: A TWO-SIDED APPROACH
FUZZY ASSOCIATION RULES: A TWO-SIDED APPROACH M. De Cock C. Cornelis E. E. Kerre Dept. of Applied Mathematics and Computer Science Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium phone: +32
More information